In order to push further estimates (REF ) and (), we need to estimate the rough integrals from (REF ), (REF ), (REF ) and (REF ). This is done by considering, for any \(m\ge 1\) , the function \((X,\mathbf {J},\mathbf {J}^{-1}, \mathcal {X}_1,\mathcal {X}_2,\ldots ,\mathcal {X}_m, \mathcal {Y}_1,\mathcal {Y}_2,\ldots ,\mathcal {Y}_m)\) 
as a solution of a (system of) RDE given by (REF )-(REF )-(REF )-(REF )-(REF )-(REF )-(REF ) driven by \(\mathbf {Z}\)  which is the lifted Gaussian process \(Z = (W,B)\)  (which satisfies Assumption REF ) in the geometric \(p\) -rough paths. It turns out that, for any \(q>2\)  large enough, we have estimate [1]}
\(\mathbb {E}\left[\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}^q\right] \lesssim _q \kappa _{a,t}.\) 
